**1. Introduction**

X-ray crystallography, NMR (Nuclear Magnetic Resonance) spectroscopy, and dual polarization interferometry, etc are indeed very powerful tools to determine the 3D structure of a protein (including the membrane protein), though they are time-consuming and costly. However, for some proteins, due to their unstable, noncrystalline and insoluble nature, these tools cannot work. Under this condition, mathematical and physical theoretical methods and computational approaches allow us to obtain a description of the protein 3D structure at a submicroscopic level. This Chapter presents some practical and useful mathematical optimization computational approaches to produce 3D structures of the Prion AGAAAAGA Amyloid Fibrils, from an energy minimization point of view.

X-ray crystallography finds the X-ray final structure of a protein, which usually need refinements in order to produce a better structure. The computational methods presented in this Chapter can be also acted as a tool for the refinements.

All neurodegenerative diseases including Parkinson's, Alzheimer's, Huntington's, and Prion's have a similarity, which is they all featured amyloid fibrils (en.wikipedia.org/wiki/Amyloid and references (Nelson et al., 2005; Sawaya et al., 2007; Sunde et al., 1997; Wormell, 1954; Gilead and Gazit, 2004; Morley et al. 2006; Gazit, 2002; Pawar et al., 2005; and references therein). A prion is a misshapen protein that acts like an infectious agent (hence the name, which comes from the words protein and infection). Prions cause a number of fatal diseases such as 'mad cow' disease in cattle, scrapie in sheep and kuru and Creutzfeldt-Jakob disease (CJD) in humans. Prion diseases (being rich in βsheets (about 43% β-sheet) (Griffith, 1967; Cappaia and Collins, 2004; Daude, 2004; Ogayar

and Snchez-Prez, 1998; Pan et al., 1993; Reilly, 2000) belong to neurodegenerative diseases. Many experimental studies such as (Brown, 2000; Brown, 2001; Brown, 1994; Cappai and Collins, 2004; Harrison et al., 2010; Holscher, 1998; Jobling et al., 2001; Jobling et al., 1999; Kuwata et al., 2003; Norstrom and Mastrianni, 2005; Wegner et al., 2002; Laganowsky et al., 2012; Jones et al., 2012; Sasaki et al., 2008; Haigh et al., 2005; Kourie et al., 2003; Zanuy et al., 2003; Kourie, 2001; Chabry et al., 1998; Gasset et al., 1992) have shown that the normal hydrophobic region (113-120) AGAAAAGA of prion proteins is an inhibitor/blocker of prion diseases. PrP lacking this palindrome could not convert to prion diseases. The presence of residues 119 and 120 (the two last residues within the motif AGAAAAGA) seems to be crucial for this inhibitory effect. The replacement of Glycine at residues 114 and 119 by Alanine led to the inability of the peptide to build fibrils but it nevertheless increased. The A117V variant is linked to the GSS disease. The physiological conditions such as pH (Cappai and Collins, 2004) and temperature (Wagoner et al., 2011) will affect the propensity to form fibrils in this region. The 3D atomic resolution structure of PrP (106-126), i.e. TNVKHVAGAAAAGAVVGGLGG, can be looked as the structure of a control peptide (Cheng et al., 2011; Lee et al., 2008). Ma and Nussinov (2002) established homology structure of AGAAAAGA and did its molecular dynamics simulation studies. Recently, Wagoner et al. computer simulation studied the structure of GAVAAAAVAG of mouse prion protein (Wagoner, 2010; Wagoner et al., 2011). Furthermore, the author computationally clarified that prion AGAAAAGA segment indeed has an amyloid fibril forming property (Fig. 1).

**Figure 1.** Prion AGAAAAGA (113-120) is surely and clearly identified as the amyloid fibril formation region, because its energy is less than the amyloid fibril formation threshold energy of -26 kcal/mol (Zhang et al., 2007).

However, to the best of the author's knowledge, there is little X-ray or NMR structural data available to date on AGAAAAGA (which falls just within the N-terminal unstructured region (1.–123) of prion proteins) due to its unstable, noncrystalline and insoluble nature. This Chapter will computationally study the molecular modeling (MM) structures of this region of prions.
